You need an invite code to join a Private contest. If you have one, you're all sorted.
If you have the link to join the contest, click on it, tap 'Join Contest' create your team, pay the entry amount and join right in.
If you have the contest code, select the match and click on the 'Cup' icon on the right-hand top corner, enter the code, tap 'Join this contest' and that's it.